A Phase transition in zero count probability for Stationary Gaussian Processes

We study the probability that a real stationary Gaussian process has at least $\eta T$ zeros in $[0,T]$ (overcrowding), or at most this number (undercrowding). We show that if the spectral measure of the process is supported on $\pm[B,A]$, overcrowding probability transitions from exponential decay to Gaussian decay at $\eta=\tfrac{A}{\pi}$, while undercrowding probability undergoes the reverse transition at $\eta=\tfrac{B}{\pi}$.Comment: 17 page